bibkey: christiana2025homflypt authors: A. Christiana; B. Clingenpeel; H. Guo; J. Oh; J. H. Przytycki; X. Wang; H. Yun year: 2025 title: “Low Dimensional Homology of the Yang-Baxter Operators Yielding the HOMFLYPT Polynomial” doi: null url: https://arxiv.org/abs/2502.20659v1 claim: “Conjecture 5.6 asserts that the short exact sequence of chain complexes for the ordered-block Künneth subcomplex splits.” strata_touched:

• D5/S3/HomologicalAlgebra/HomflyptYangBaxterKunnethSplittingRefutation license: citation-only triage: anchor

HOMFLYPT Yang-Baxter homology and the Künneth subcomplex

Verified locator

https://arxiv.org/abs/2502.20659v1 — arXiv version 1, §5.1, Definition 5.4, Proposition 5.5 and Conjecture 5.6. The normalized operator and traveller face maps are Definitions 1.5 and 1.7 of this version. The title and authors agree with the arXiv record.

Mathematical scope

A. Christiana, B. Clingenpeel, H. Guo, J. Oh, J. H. Przytycki, X. Wang, H. Yun, arXiv:2502.20659v1, “Low Dimensional Homology of the Yang-Baxter Operators Yielding the HOMFLYPT Polynomial”.

The normalized operator is Definition 1.5, pp. 3–4 of the v1 PDF. Its coefficients are $1$ if $d=a\ge b=c$, $y^2$ if $d=a<b=c$, $1-y^2$ if $c=a<b=d$, and zero otherwise. The coefficient ring is $\mathbb{Z}[y^2]$. Writing $t=y^2$ identifies this ring with Polynomial ℤ. On a basis pair, $R(a\otimes b)=b\otimes a$ for $a\ge b$ and $R(a\otimes b)=(1-t)a\otimes b+tb\otimes a$ for $a<b$.

Definition 1.7, p. 4, describes the face maps:

More precisely, whenever we see a crossing, we apply the Yang-Baxter operator, and for straight lines, we apply the identity map. When hitting the left wall, we delete the first tensor element of each basis in the linear combination, and when hitting the right wall, we delete the last tensor element of each basis in the linear combination.

The boundary is ${\partial }_n={\sum }_{i=1}^n(-1{)}^i(d_i^l-d_i^r)$. Both face maps use the crossing shown in Figure 1.1; the traveller is the left output when travelling left and the right output when travelling right.

Definition 5.4, p. 18:

Fix a decomposition of letters $X_{(m)}=A\cup B$ and consider a submodule $C_n^{(m,A,B)}$ of $C_n^m$ generated by sequences in $X_{(m)}^n$ which start with letters from $A$ and end with letters in $B$. More formally:

$X_{(n,m,A,B)}={(a_1,a_2,…,a_n)\mid\text{there is }i\text{ such that } a_1,…,a_i\in A;\ a_{i+1},…,a_n\in B}.$

Every cutoff $i\in \{0,\dots ,n\}$ is included. Proposition 5.5, p. 18, assumes $A\ge B$, meaning $v_i\in A$ and $v_j\in B$ imply $i\ge j$. It states that this is a subchain complex and gives the concatenation-boundary formula. Its tensor-product assertion additionally assumes $A>B$.

Conjecture 5.6, p. 18:

The short exact sequence of chain complexes

$$0\to C_{\bullet }^{(m,A,B)}\to C_{\bullet }^m\to C_n^m/C_{\bullet }^{(m,A,B)}\to 0,$$

splits. Thus homology $H_n(C_{\bullet }^m)$ splits.

A splitting here is a chain-map retraction of the inclusion. A retraction that is merely linear in each degree does not imply the stated homology consequence. The formal encoding writes its chain-map equality in the ambient module through the injective subtype map; this is the equality ${\iota }_n{\partial }_{n+1}^D{p}_{n+1}={\iota }_n{p}_n{\partial }_{n+1}$.

The formal refutation uses $m=2$, $A=\{2\}$ and $B=\{1\}$, encoded by Fin 2 as $1$ and $0$. In degree five each of the six block generators has zero boundary. The chain

$$c=t(t+1)11112+(t^3+t^2+t+1)11121+t11212-t12112+t^{3}12211+t^{2}12221$$

has boundary $t^3(t^2-1)(2221-2111)$, a nonzero element of the block submodule in degree four. Any chain retraction would fix this boundary and also send it to zero, which is impossible. This refutes the asserted chain splitting; it does not by itself refute every possible abstract isomorphism between homology groups, nor Proposition 5.5.